Superimposition

Antigoni Kaliontzopoulou, CIBIO/InBIO, University of Porto

14 October, 2019

What is Shape?

What is Shape?

Scale and Size

Note: one can perform GM analyses without recording digitizing ‘scale’, but in this case one has no inherent size estimate, and no notion of its effect on shape

Size in GM Studies

Size in GM Studies

\[\small{CS}=\sqrt{\sum_{i,j}^{k,p}\left(\mathbf{Y}_{ij}-\mathbf{Y}_{ic}\right)^2}\]

where \(\small{p}\) is the number of landmarks and \(\small{k}\) is the number of coordinate dimensions

Centroid Size (CS): Properties

Shape Differences

Shape Differences

\[\small{D}_{Proc}=\sqrt{\sum_{i,j}^{k,p}\left(\mathbf{Y}_{1.ij}-\mathbf{Y}_{2.ij}\right)^2}\]

Shape from Landmarks

Shape from Landmarks

Shape from Landmarks

\(\small\mathbf{Z}=\frac{1}{CS}\mathbf{(Y-\overline{Y})H}\)

Procrustes Superimposition: Translation

Procrustes Superimposition: Scale

\[\small{CS}=\sqrt{\sum_{i,j}^{k,p}\left(\mathbf{Y}_{ij}-\mathbf{Y}_{ic}\right)^2}=\]

\[\small{[tr}[\mathbf{(Y-\overline{Y})(Y-\overline{Y})^T}]]^{-1/2}=1\]

Procrustes Superimposition: Rotation

How Much to Rotate?

\(\small\mathbf{Z}=\frac{1}{CS}\mathbf{(Y-\overline{Y})H}\)

How Much to Rotate?

\(\small\mathbf{Z}=\frac{1}{CS}\mathbf{(Y-\overline{Y})H}\)

How Much to Rotate?

\(\small\mathbf{Z}=\frac{1}{CS}\mathbf{(Y-\overline{Y})H}\)

Ordinary Procrustes Analysis: Review

\(\small\mathbf{Z}=\frac{1}{CS}\mathbf{(Y-\overline{Y})H}\)

Generalized Procrustes Analysis

Generalized Procrustes Analysis

GPA: Example

GPA: Example (Cont.)

GPA: Step By Step

GPA: Group Differences

##           Df       SS        MS     Rsq      F      Z Pr(>F)   
## gp         1 0.009180 0.0091801 0.09059 5.8772 4.5952  0.001 **
## Residuals 59 0.092158 0.0015620 0.90941                        
## Total     60 0.101338                                          
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

TPS of male vs. female Podarcis (10X magnification)

GPA: Exploring Shape Variation

##  [1] 0.019120833 0.014962558 0.013953399 0.011872076 0.011252644
##  [6] 0.010428595 0.010172139 0.008714636 0.008158964 0.007586907
## [11] 0.006831716 0.006049213 0.005992073 0.005275534 0.004900721
## [16] 0.004129789 0.003339428 0.003200212 0.002954954 0.002591265
## [21] 0.000000000 0.000000000 0.000000000 0.000000000

GPA: Data Dimensionality

GPA: Data Dimensionality

GPA: Data Dimensionality

NOTE: as we’ll see later, use of permutation methods (RRPP) alleviates this issue.

GPA: Extensions to Three Dimensions

Now we have: \(\tiny{Y}=\begin{bmatrix} X_1 & Y_1 & Z_1 \\ X_2 & Y_2 & Z_2 \\ \vdots & \vdots & \vdots \\ X_p & Y_p & Z_p \\ \end{bmatrix}\) & \(\small\mathbf{Z}=\frac{1}{CS}\mathbf{(Y-\overline{Y})H}\)

GPA: Extensions to Three Dimensions

Now we have: \(\tiny{Y}=\begin{bmatrix} X_1 & Y_1 & Z_1 \\ X_2 & Y_2 & Z_2 \\ \vdots & \vdots & \vdots \\ X_p & Y_p & Z_p \\ \end{bmatrix}\) & \(\small\mathbf{Z}=\frac{1}{CS}\mathbf{(Y-\overline{Y})H}\)

Modifications: Full Vs. Partial Procrustes Fitting

Historical Note: Bookstein’s Shape Coordinates

Historical Note: Bookstein’s Shape Coordinates

Bookstein’s Shape Coordinates

Modifications: Resistant-Fit

Resistant-Fit Superimposition

Resistant-Fit: Comments

Superimposition: Summary

Shape Spaces from GPA

\[\tiny{D}_{Proc}=\sqrt{\sum_{i,j}^{k,p}\left(\mathbf{Y}_{1.ij}-\mathbf{Y}_{2.ij}\right)^2}\]

Example with 2,000 random (uniform) triangles. Note overabundance of shapes near ‘north pole’.

Tangent Space

Tangent Space